hypergraphs, metabolic networks, bioreaction systems.trelat/gdt/confs/georges...computation of...
TRANSCRIPT
Hypergraphs, Metabolic Networks, Bioreaction Systems.
G. Bastin
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PART 1 : Metabolic flux analysis and minimal bioreaction modelling
PART 2 : Dynamic metabolic flux analysis of underdetermined networks
PART 1 : Metabolic flux analysis and minimal bioreaction modelling
CELLSSubstrates
ProductsLactate
AlanineGlucose
CO2Glutamine NH4
Growth of CHO cells in a serum-free medium
The basic issue
CELLSSubstrates
ProductsLactate
AlanineGlucose
CO2Glutamine NH4
Growth of CHO cells in a serum-free medium
Question : What is the minimal set of input-output bioreactions ?
• consistent with cell metabolism • explaining measurements in the culture medium
Outline
1. Metabolic network and stoichiometric matrix
2. Convex basis and elementary pathways
3. From elementary pathways to bioreactions
4. Experimental data and metabolic flux analysis
5. Computation of minimal sets of bioreactions
6. Final remarks
Metabolic network
inputs
outputs
intracellular intermediate metabolites
Intracellular biochemical reaction
inputs
outputs
intracellular intermediate metabolites
Remark 1 : for the sake of clarity, this is a simplified example representing only the metabolism of « energetic substrates » Glucose and Glutamine. Metabolism of amino-acids is not considered.
But the methodology applies to more complex networks as we shall see in Part II.
Metabolic network
inputs
outputs
intracellular intermediate metabolites
Remark 2 : Only internal nodes that are at « branching points » (no loss of generality).
Metabolic network
Glycolysis
metabolic flux
stoichiometric coefficient
Metabolic network
Stoichiometric matrix N
1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1
Glucose 6-P Dihydro-Ac-3P
Ribose-5-P Glyc-3-P Pyruvate
Acetyl-coA Citrate
Alpha-ketoglut Fumarate
Malate Oxaloacetate
Aspartate Glutamate
CO2
(= incidence matrix of the network)
internal nodes
Metabolic fluxes
Stoichiometric matrix N
1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1
Glucose 6-P Dihydro-Ac-3P
Ribose-5-P Glyc-3-P Pyruvate
Acetyl-coA Citrate
a-ketoglutarate Fumarate
Malate Oxaloacetate
Aspartate Glutamate
CO2
(= incidence matrix of the network)
1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1
Stoichiometric matrix N
The set of non-negative vectors of the kernel of N is a polyhedral cone in the non-negative orthant
Convex basis1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1
Stoichiometric matrix
The convex basis is the set of
edges of the cone
matrix E
Elementary pathway Metabolic
interpretation of convex basis
Remark 3. In the literature, « Elementary pathways » are also called : « Elementary (flux) modes » or « Extreme pathways ».
Elementary pathway
C3H6O3C6H12O6 �! 2
Elementary pathway
Another example …
From Convex basis to Bioreaction system
CELLSSubstrates
ProductsLactate
AlanineGlucose
CO2Glutamine NH4
Nucleotides
Experimental data(batch)
Glucose
Lactate
NH4
Glutamine
Alanine
Cell density
Glucose
Lactate
NH4
Glutamine
Alanine
Green dots = exponential growth
Cell density
Experimental data(batch)
Cell density
Computation of specific consumption and production rates by linear regression
Glucose Lactate
NH4
Glutamine
Alanine
Metabolic flux analysis
Metabolic flux analysis
Non-negative decomposition of in the convex basis
Metabolic flux analysis
The 12 equivalent minimal decompositions
Exactly five non-zero coefficients in each vector !
12 equivalent minimal (sub)sets of bioreactions
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Density Glucose
LactateGlutamine
Alanine Ammonia
PART 2 : Dynamic metabolic flux analysis of underdetermined
networks
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Case study : Hybridoma cells for production
of Immunoglobuline
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CELLSSubstratesProductsLactate
AlanineGlucose
CO2Glutamine
NH4
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CELLSSubstratesProductsLactate
AlanineGlucose
CO2Glutamine
NH4
Glutamate Serine
Arginine Asparagine
Aspartate Histidine
Leucine Isoleucine
Lysine Methionine
Phenylalanine Threonine
Tryptophan Valine
Tyrosine (Proline)
(Cysteine)
Glycine Immunoglobuline
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Central Metabolism
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Amino-Acids Metabolism
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Antibody Synthesis
Biomass Synthesis
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Perfusion culture
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dX
dt= µX � ↵DX
dS
dt= ��SX +D(Sin � S)
dP
dt= �PX �DP
biomass
substrate
product
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DATA batch perfusion
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r = 70 reactions n = 44 internal metabolites p = 22 measurements
Underdetermined system !
Metabolic Flux Analysis ProblemN�(t) = 0
Nm�(t) = �m(t)
0 �(t)
n x r
p x r
dX
dt= µX � ↵DX
dS
dt= ��SX +D(Sin � S)
dP
dt= �PX �DP
biomass
substrate
product
✓��S�P
◆= �m
N�(t) = 0
Nm�(t) = �m(t)
0 �(t)
Again the set of solutions is a pointed polyhedral cone in the positive orthant
� =qX
i=1
!ifi⇣!i � 0,
qX
i=1
!i = 1⌘
edges of the cone
N�(t) = 0
Nm�(t) = �m(t)
0 �(t)
Again the set of solutions is a pointed polyhedral cone in the positive orthant
� =qX
i=1
!ifi⇣!i � 0,
qX
i=1
!i = 1⌘
edges of the cone
� =
0
BBB@
�1�2...�r
1
CCCA
Solution intervals
�min
i �i �max
i
�min
i = min
�fki, k = 1, . . . , q
,
�max
i = max
�fki, k = 1, . . . , q
Results : Glycolysis fluxes
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Results :TCA-cycle fluxes
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Final remarks
2. « Minimal » must be understood as « data compression »: there is no loss of information in the « minimal model » with respect to the initial metabolic network.
3. Combinatorial explosion. In « Zamorano et al., an example of CHO cells where the network involves 84930 elementary pathways and leads to 88926 equivalent minimal models ! …. but each minimal model includes only 22 bioreactions and the polyhedral cone for the interval metabolic flux analysis has only 32 edges.
Combinatorial explosion of the number of elementary pathways is not an issue because these 22 bioreactions and 32 edges can be computed directly !
1. Dynamical model under pseudo-steady-state assumption of metabolic flux analysis = order reduction by singular perturbation.
References
A. Provost, G. Bastin, "Dynamical metabolic modelling under the balanced growth condition", Journal of Process Control, Vol. 14(7), 2004, pp. 717 - 728.
F. Zamorano, A. Vande Wouwer, G. Bastin, "A detailed metabolic flux analysis of an undetermined network of CHO cells", Journal of Biotechnology, Vol.150(4), 2010, pp. 497 - 508.
R.M. Jungers, F. Zamorano, V.D. Blondel, A. Vande Wouwer, G. Bastin, "Fast computation of minimal elementary decompositions of metabolic flux vectors", Automatica, Special issue on Systems Biology, Vol. 47(7), 2011, pp. 1255 - 1259.
S. Fernandes de Souza, G.Bastin, M. Jolicoeur, A. Vande Wouwer, "Dynamic metabolic flux analysis using a convex analysis approach: application to hybridoma cell cultures in perfusion", Biotechnology and Bioengineering, 2016, in press.
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Thank You !
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